### Genus 2 function fields with degree 2 elliptic subfields., by Tony Shaska

We study genus 2 function fields with elliptic subfields of degree 2. The locus $\L_2$ of these fields is a 2-dimensional subvariety of the moduli space $\mathcal M_2$ of genus 2 fields. An equation for $\L_2$ is already in the work of Clebsch and Bolza. We use a birational parametrization of $\L_2$ by affine 2-space to study the relation between the j-invariants of the degree 2 elliptic subfields. This extends work of Geyer, Lange, Stichtenoth and others. We find a 1-dimensional family of genus 2 curves having exactly two isomorphic elliptic subfields of degree 2; this family is parametrized by the j-invariant of these subfields.